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\topic{Lecture 4 \\Differential Calculus-I\\ \scriptsize Partial Differentiation (22 Sep 2009)}
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\subsection*{Functions of two or more variables}
Consider the volume of a right circular cylinder of radius $r$ and height $h$,  \[V = \pi r^2h\]
which clearly implies that the volume $V$ has definite value for a given pair of $r, h$. Here we call $r$ and $h$, independent and $V$ as dependent variable.
\subsection*{Partial Derivative} Let $z=f(x, y)$ be function of two independent variables $x, y$. The partial differential coefficient of $f(x,y)$ with respect to $x$ is the ordinary differential coefficient of $f(x,y)$ when $y$ is regarded as a constant. We write it as
\[\frac{\partial f}{\partial x},~~or~~ f_x\]
Thus
\[f_x= \frac{\partial f}{\partial x}= \lim_{h\rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h} \]
Similarly, the partial derivative with respect to $y$ is computed as ordinary derivative, while treating $x$ as constant, i.e.,
\[f_y= \frac{\partial f}{\partial y}= \lim_{k\rightarrow 0} \frac{f(x,y+k)-f(x,y)}{k} \]
Further, higher order derivatives also can be computed. For example further derivatives of $f_x$ and $f_y$ are
\[ f_{xx},~f_{xy},~f_{yx},~f_{yy}\]
i.e., 
\[\frac{\partial^2 f}{\partial x^2},~\frac{\partial^2 f}{\partial y \partial x},~\frac{\partial^2 f}{\partial x \partial y},~\frac{\partial^2 f}{\partial y^2},~\]
It should be specially noted that 
\[\frac{\partial^2 f}{\partial y \partial x}=\frac{\partial }{\partial y} \left(\frac{\partial f}{\partial x}\right)\]
and 
\[\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial }{\partial x} \left(\frac{\partial f}{\partial y}\right)\]
But In ordinary cases
\[\frac{\partial^2 f}{\partial y \partial x}=\frac{\partial^2 f}{\partial x \partial y}\]

\begin{example}
Find the partial derivatives of $x^4+x^2y^2+y^4$
\end{example}
Let $f=x^4+x^2y^2+y^4$, Then \\
\[f_x = 4x^3+2xy^2, ~~f_y = 2x^2y+4y^3\]
\[f_{xx} = 12x^2+2y^2, ~~f_{yy} = 2x^2+12y^2, ~~f_{xy} = 4xy,  ~~f_{yx} = 4xy\]
From here also have
\[f_{xy} = f_{yx}\]

\begin{example}
Prove that $y = f (x + at) + g (x - at)$ satisfies $\frac{\partial ^{2} y}{\partial t^{2} } =a^{2} \left(\frac{\partial ^{2} y}{\partial x^{2} } \right)$.
\end{example}
Given $y&=& f (x + at) + g (x - at)$
Differentiate partially wrt $x$,
\[
\begin{array}{rcl}
  \pd yx &=& f' (x + at) + g' (x - at)\\
 \pd {^2y}{x^2} &=& f'' (x + at) + g'' (x - at)
\end{array}
\]

Differentiate partially wrt $t$,
\[
\begin{array}{rcl}
  \pd yt &=& a.f' (x + at) + (-a) g' (x - at)\\
 \pd {^2y}{t^2} &=& a^2.f'' (x + at) + (-a)^2.g'' (x - at)
\end{array}
\]

Now,
\[ \pd {^2y}{t^2} &=& a^2.f'' (x + at) + (-a)^2.g'' (x - at)=a^2[f'' (x + at) + g'' (x - at)] = a^2  \pd {^2y}{x^2}\]
\section*{Problems}
\begin{enumerate}
\item  Find $f_x$ and $f_y$, when \\
(a) $f=\tan^{-1}\frac{x^2+y^2}{x+y}$~~(b) $f=\frac{x^2}{a^2}+\frac{y^2}{b^2}-1$ ~~(c) $f=x^y$ ~~(d) $f=\log (x^2+y^2)$

\item   If $x = r \cos \theta$ , $y = r \sin \theta $, find (i) $\left(\frac{\partial x}{\partial r} \right)_{\theta } $(ii) $\left(\frac{\partial y}{\partial \theta } \right)_{r} $(iii) $\left(\frac{\partial r}{\partial x} \right)_{y} $(iv) $\left(\frac{\partial \theta }{\partial y} \right)_{x} $ 

\item   Find $\frac{\partial u}{\partial r} $ and $\frac{\partial u}{\partial \theta }$ if $u = e^{r \cos \theta} cos(r sin \theta)$ 

\item   If $z(x+y) = x^2 + y^2$, show that \[\left(\frac{\partial z}{\partial x} -\frac{\partial z}{\partial y} \right)^{2} =4\left(1-\frac{\partial z}{\partial x} -\frac{\partial z}{\partial y} \right)\]

\item   If $u = \sin^{-1}(\frac{x}{y}) + \tan^{-1}(\frac{y}{x})$, then find the value of $\left(x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} \right)$. 

\item   If $u = (1 -2xy + y^2)^{-1/2}$ prove that, $x\frac{\partial u}{\partial x}  - y\frac{\partial u}{\partial y}  = y^2u^3$.

\item   If $z = e^{ax+by}.f (ax - by)$, prove that $b\frac{\partial z}{\partial x} - a\frac{\partial z}{\partial y}  = 2abz$.

\item   If $z = x^2 \tan^{-1}\left(\frac{y}{x}\right)- y^2 \tan^{-1}\left(\frac{x}{y}\right)$, prove that$\frac{\partial ^{2} z}{\partial y\partial x} =\frac{x^{2} -y^{2} }{x^{2} +y^{2} } $.

\item   If $u = e^{xyz}$,, Show that \[\frac{\partial ^{3} z}{\partial x\partial y\partial z}=(1+3xyz+x^2y^2z^2)e^{xyz} \]

\item   If $v = (x^2 + y^2 + z^2)^{m/2}$, then find the value of $m$ ($m \ne 0$) such that \[\frac{\partial ^{2} v}{\partial x^{2} } +\frac{\partial ^{2} v}{\partial y^{2} } +\frac{\partial ^{2} v}{\partial z^{2} } =0\]

\item   If $v = (1 - xy + y^2)^{-1/2}$, prove that \[\frac{\partial }{\partial x} \left[\left(1-x^{2} \right)\frac{\partial u}{\partial x} \right]+\frac{\partial }{\partial y} \left(y^{2} \frac{\partial u}{\partial y} \right)=0\]

\item   If $f(x, y) =\frac{1}{\sqrt{y} } e^{-\frac{(x-a)^{2} }{4y} }$, Prove $f_{xy}=f_{yx}$

\item   If $z = (x + y) + (x + y)\phi \left(\frac{y}{x}\right)$, then prove that  \[x\left(\frac{\partial ^{2} z}{\partial x^{2} } -\frac{\partial ^{2} z}{\partial y\partial x} \right) =y\left(\frac{\partial ^{2} z}{\partial y^{2} } -\frac{\partial ^{2} z}{\partial x\partial y} \right)\]

\item   If $u = x^y$, show that \[\frac{\partial ^{3} u}{\partial x^{2} \partial y} =\frac{\partial ^{3} u}{\partial x\partial y\partial x}\]

\item   If $u = \log (x^3 + y^3 + z^3 + 3xyz)$, show that 
\[\left(\frac{\partial }{\partial x} +\frac{\partial }{\partial y} +\frac{\partial }{\partial z} \right)^{2} u=-\frac{9}{\left(x+y+z\right)^{2} }\]

\item   If \[\frac{x^{2} }{a^{2} +u} +\frac{y^{2} }{b^{2} +u} +\frac{z^{2} }{c^{2} +u} =1\] Show that: \[\left(\frac{\partial u}{\partial x} \right)^{2} +\left(\frac{\partial u}{\partial y} \right)^{2} +\left(\frac{\partial u}{\partial z} \right)^{2} =2\left(x\frac{\partial u}{\partial x} +y\frac{\partial u}{\partial y} +z\frac{\partial u}{\partial z} \right)\]

\item  If $x^xy^yz^z = c$, Show that at $x=y=z$, \[\frac{\partial^2z}{\partial x \partial y} = -(x \log ex)^{-1}\]
\end{enumerate}

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